YES

# Compositional parallel critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  g(f(a())) -> f(g(f(a())))
  g(f(a())) -> f(f(a()))
  f(f(a())) -> f(a())

Let C be the following subset of R:

  g(f(a())) -> f(g(f(a())))
  g(f(a())) -> f(f(a()))
  f(f(a())) -> f(a())

The parallel critical pair system PCPS(R,C) is:

  (empty)

All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

  g(f(a())) -> f(g(f(a())))
  g(f(a())) -> f(f(a()))
  f(f(a())) -> f(a())

All parallel critical peaks (except C's) are decreasing wrt rule labeling:

  phi(g(f(a())) -> f(g(f(a())))) = 3
  phi(g(f(a())) -> f(f(a()))) = 1
  phi(f(f(a())) -> f(a())) = 2

  psi(g(f(a())) -> f(g(f(a())))) = 3
  psi(g(f(a())) -> f(f(a()))) = 2
  psi(f(f(a())) -> f(a())) = 1